18 research outputs found
Duality of equations and coequations via contravariant adjunctions
In this paper we show duality results between categories of equations and categories of coequations. These dualities are obtained as restrictions of dualities between categories of algebras and coalgebras, which arise by lifting contravariant adjunctions on the base categories. By extending this approach to (co)algebras for (co)monads, we retrieve th
Minimisation in Logical Form
Stone-type dualities provide a powerful mathematical framework for studying
properties of logical systems. They have recently been fruitfully explored in
understanding minimisation of various types of automata. In Bezhanishvili et
al. (2012), a dual equivalence between a category of coalgebras and a category
of algebras was used to explain minimisation. The algebraic semantics is dual
to a coalgebraic semantics in which logical equivalence coincides with trace
equivalence. It follows that maximal quotients of coalgebras correspond to
minimal subobjects of algebras. Examples include partially observable
deterministic finite automata, linear weighted automata viewed as coalgebras
over finite-dimensional vector spaces, and belief automata, which are
coalgebras on compact Hausdorff spaces. In Bonchi et al. (2014), Brzozowski's
double-reversal minimisation algorithm for deterministic finite automata was
described categorically and its correctness explained via the duality between
reachability and observability. This work includes generalisations of
Brzozowski's algorithm to Moore and weighted automata over commutative
semirings.
In this paper we propose a general categorical framework within which such
minimisation algorithms can be understood. The goal is to provide a unifying
perspective based on duality. Our framework consists of a stack of three
interconnected adjunctions: a base dual adjunction that can be lifted to a dual
adjunction between coalgebras and algebras and also to a dual adjunction
between automata. The approach provides an abstract understanding of
reachability and observability. We illustrate the general framework on range of
concrete examples, including deterministic Kripke frames, weighted automata,
topological automata (belief automata), and alternating automata
An Eilenberg-like theorem for algebras on a monad
An Eilenberg–like theorem is shown for algebras on a given monad. The main idea is to explore the
approach given by Bojan´czyk that defines, for a given monad T on a category D, pseudovarieties of
T–algebras as classes of finite T–algebras closed under homomorphic images, subalgebras, and finite
products. To define pseudovarieties of recognizable languages, which is the other main concept for
an Eilenberg–like theorem, we use a category C that is dual to D and a recent duality result
between Eilenberg–Moore categories of algebras and coalgebras by Salamanca, Bonsangue, and Rot.
Using this duality we define the concept of a pseudovariety o
How to write a coequation
There is a large amount of literature on the topic of covarieties,
coequations and coequational specifications, dating back to the early
seventies. Nevertheless, coequations have not (yet) emerged as an everyday
practical specification formalism for computer scientists. In this review
paper, we argue that this is partly due to the multitude of syntaxes for
writing down coequations, which seems to have led to some confusion about what
coequations are and what they are for. By surveying the literature, we identify
four types of syntaxes: coequations-as-corelations, coequations-as-predicates,
coequations-as-equations, and coequations-as-modal-formulas. We present each of
these in a tutorial fashion, relate them to each other, and discuss their
respective uses
Guarded Kleene Algebra with Tests: Coequations, Coinduction, and Completeness
Guarded Kleene Algebra with Tests (GKAT) is an efficient fragment of KAT, as it allows for almost linear decidability of equivalence. In this paper, we study the (co)algebraic properties of GKAT. Our initial focus is on the fragment that can distinguish between unsuccessful programs performing different actions, by omitting the so-called early termination axiom. We develop an operational (coalgebraic) and denotational (algebraic) semantics and show that they coincide. We then characterize the behaviors of GKAT expressions in this semantics, leading to a coequation that captures the covariety of automata corresponding to these behaviors. Finally, we prove that the axioms of the reduced fragment are sound and complete w.r.t. the semantics, and then build on this result to recover a semantics that is sound and complete w.r.t. the full set of axioms
Guarded Kleene Algebra with Tests: Coequations, Coinduction, and Completeness
Guarded Kleene Algebra with Tests (GKAT) is an efficient fragment of KAT, as it allows for almost linear decidability of equivalence. In this paper, we study the (co)algebraic properties of GKAT. Our initial focus is on the fragment that can distinguish between unsuccessful programs performing different actions, by omitting the so-called early termination axiom. We develop an operational (coalgebraic) and denotational (algebraic) semantics and show that they coincide. We then characterize the behaviors of GKAT expressions in this semantics, leading to a coequation that captures the covariety of automata corresponding to these behaviors. Finally, we prove that the axioms of the reduced fragment are sound and complete w.r.t. the semantics, and then build on this result to recover a semantics that is sound and complete w.r.t. the full set of axioms